where \({\hat{s}}\)  is the output of the pre-trained sequential encoder that takes as input \({\hat{v}}_{i_1}, \ldots , {\hat{v}}_{i_t}\) , and the item representations \({\hat{v}}_i\)  can be derived based on the code \({\hat{c}}_{i}\)  in a similar way in Eqn. (REF ).
To make the permutation matrices \({\Pi }_k\)  differentiable, we are inspired by Birkhoff's theorem [1]}: any doubly stochastic matrix can be considered as a convex combination of permutation matrices. Following this idea, we use doubly stochastic matrices to simulate the permutation. Concretely, we start by randomly initializing a parameter matrix for each dimension \({\Theta }_k \in \mathbb {R}^{M\times M}\) , then convert it into the doubly stochastic matrix via Gumbel-Sinkhorn algorithm [2]}.
Then, we optimize \(\lbrace {\Theta }_1, \ldots , {\Theta }_D\rbrace \)  according to the next item probability (Eqn. (REF )) using cross-entropy loss, while fixing the rest parameters in the pre-trained recommender.
